As the name suggests, the mean or average of a function in BMO does not oscillate very much when computing it over cubes close to each other in position and scale. Precisely, if Q and R are dyadic cubes such that their boundaries touch and the side length of Q is no less than one-half the side length of R (and vice versa), then

|fR−fQ|≤C‖f‖BMO{\displaystyle |f_{R}-f_{Q}|\leq C\|f\|_{BMO}}

where C > 0 is some universal constant. This property is, in fact, equivalent to f being in BMO, that is, if f is a locally integrable function such that |fR−fQ| ≤ C for all dyadic cubes Q and R adjacent in the sense described above and f is in dyadic BMO (where the supremum is only taken over dyadic cubes Q), then f is in BMO[7].

The John–Nirenberg inequality implies that A(f) ≤ C||f||BMO for some universal constant C. For an L∞ function, however, the above inequality will hold for all A > 0. In other words, A(f) = 0 if f is in L∞. Hence the constant A(f) gives us a way of measuring how far a function in BMO is from the subspace L∞. This statement can be made more precise:[9] there is a constant C, depending only on the dimensionn, such that for any function f ∈ BMO(Rn) the following two-sided inequality holds

Definition 3. An Analytic function on the unit disk is said to belong to the Harmonic BMO or in the BMOH space if and only if it is the Poisson integral of a BMO(T) function. Therefore, BMOH is the space of all functions u with the form:

Charles Fefferman in his original work proved that the real BMO space is dual to the real valued harmonic Hardy space on the upper half-spaceRn × (0, ∞].[11] In the theory of Complex and Harmonic analysis on the unit disk, his result is stated as follows.[12] Let Hp(D) be the Analytic Hardy space on the unit Disc. For p = 1 we identify (H1)* with BMOA by pairing f ∈ H1(D) and g ∈ BMOA using the anti-linear transformationTg

Notice that although the limit always exists for an H1 function f and Tg is an element of the dual space (H1)*, since the transformation is anti-linear, we don't have an isometric isomorphism between (H1)* and BMOA. However one can obtain an isometry if they consider a kind of space of conjugate BMOA functions.

The space VMO of functions of vanishing mean oscillation is the closure in BMO of the continuous functions that vanish at infinity. It can also be defined as the space of functions whose "mean oscillations" on cubes Q are not only bounded, but also tend to zero uniformly as the radius of the cube Q tends to 0 or ∞. The space VMO is a sort of Hardy space analogue of the space of continuous functions vanishing at infinity, and in particular the real valued harmonic Hardy space H1 is the dual of VMO.[13]

Let Δ denote the set of dyadic cubes in Rn. The space dyadic BMO, written BMOd is the space of functions satisfying the same inequality as for BMO functions, only that the supremum is over all dyadic cubes. This supremum is sometimes denoted ||•||BMOd.

This space properly contains BMO. In particular, the function log(x)χ[0,∞) is a function that is in dyadic BMO but not in BMO. However, if a function f is such that ||f(•−x)||BMOd ≤ C for all x in Rn for some C > 0, then by the one-third trickf is also in BMO.

Although dyadic BMO is a much narrower class than BMO, many theorems that are true for BMO are much simpler to prove for dyadic BMO, and in some cases one can recover the original BMO theorems by proving them first in the special dyadic case.[15]

^Aside with the collected papers of Fritz John, a general reference for the theory of functions of bounded mean oscillation, with also many (short) historical notes, is the noted book by Stein (1993, chapter IV).